MAML, Arithmetic with * Operations: Arithmetic Patterns
This is a MAML (Maine Association of Math Leagues) problem from October 2010 (Meet 1 of 2010-11), Round 1, Problem 3: http://maml.net/problems.htm
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Condensed Written Solution:
First, re-write the pattern as the following:
(1000-1)^2=1000(1000-2)+1^2
(1000-2)^2=1000(1000-4)+2^2
(1000-3)^2=1000(1000-6)+3^2
From this, we get the following formula:
(1000-n)^2=1000(1000-2n)+n^2
Now, notice that:
994=1000-6
989=1000-11
Therefore, we can plug them into our formula:
994^2=(1000-6)^2=1000(1000-2(6))+6^2=988,036
989^2=(1000-11)^2=1000(1000-2(11))+11^2=978,121
Add the two numbers to find our answer:
988,036+978,121=1,966,157 ANSWER